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6. Truth and Possible Worlds
We have defined logical entailment, consistency, and the connectives
∧
,
∨
,
¬
, all in terms of
belief.
In view of the close connection between belief and truth, described in the first section, we
should see what each of these has to do with truth.
Suppose
A
is true, and
A
entails
B
.
Does
B
then have to be true as well, or could it be false?
If it
is possible for
B
to be false here, then that’s bad news.
For then even a perfect thinker, starting
with truth, could end up in error.
Fortunately we shall see that this is not possible.
Logical
entailment cannot lead from truth to something false.
Entailment, we say, is
truth preserving
.
6.1 Possible Worlds
There are many other links between the logical terms and truth.
In order to investigate all of
these it will be useful to introduce the notion of a
possible world
.
This will allow to define the
structure of propositions more precisely than before, and see just what it means for a proposition
to be true.
Recall the idea of an
expansion
of an epistemic state – knowledge is added to the state, without
removing any.
This notion of expansion naturally leads to the idea of a
learning path
of
epistemic states.
A learning path is a sequence of epistemic states where each state is an
expansion of the previous one.
A learning path might be seen as the intellectual biography of a
perfect thinker, as she acquires more and more knowledge.
One question about learning paths is whether they can ever come to an end.
Is there any
epistemic state that’s so big it can’t be expanded further?
Theoretically, at least, a perfect thinker
might get to the stage where she has a firm, definite opinion on every issue.
She has unshakable
beliefs about the doings of every single beetle in the Amazon basin, knows everything there is to
know about every person on the planet, even the number of hairs on their head.
Her epistemic
state is every bit as detailed and complicated as the real world itself!
No human could have such
an epistemic state, of course; it’s just a theoretical entity.
Such an epistemic state can be called
maximal
, since it cannot be expanded.
It is not contained
in any epistemic state (apart from itself of course).
We will call such a maximal epistemic state a
possible world
.
Definition
A
possible world
is a maximal epistemic state, i.e. one which cannot be expanded.
It is not contained in any other epistemic state.
There are many possible worlds, in this sense, as there are many possible combinations of
beliefs.
A possible world doesn’t have to match the real world, it only has to be a possible state
of belief for a perfect thinker.
It has to be coherent, make sense, and so on, but it doesn’t have to
be true.
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The opposite of a maximal epistemic state (possible world) is a
minimal
epistemic state.
This is
a state with no information at all, so that it is contained in every epistemic state, even itself.
It is
clear that there can only be one minimal state.
(Why?)
Assuming it exists, we will call this state
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 Winter '09
 Burkholder,Leslie
 Logic, Boolean Algebra, Logical connective, Propositional calculus, epistemic state

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